Transformed statistical distance measures and the fisher information matrix

• Published in: Linear algebra and its applications Vol. 437; no. 2; pp. 692 - 712
• Main Authors: Klein, André, Spreij, Peter
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.07.2012; Elsevier
• Subjects: Algebra; Applied sciences; Euclidean distance; Exact sciences and technology; Fisher information matrix; Givens rotation matrix; Information theory; Information, signal and communications theory; Linear and multilinear algebra, matrix theory; Mathematics; Quantum information; Sciences and techniques of general use; Statistical distance measures; Sylvester resultant matrix; Telecommunications and information theory; 15A23; 15B99; Givens rotation matrix; Euclidean distance; 62H30; Sylvester resultant matrix; Quantum information; 15A18; Statistical distance measures; Fisher information matrix; Singularity; Variability; Matrix factorization; Coordinate system; Multivariate analysis; Relevance; Information measure; Covariance; Semantics; Measurement procedure; Eigenvalue problem; Stationary process; Rotation angle; Euclidean geometry; Statistical analysis; Probabilistic approach; Conceptual analysis; Resultant matrix; Fisher information; Discrete geometry; Statistical parameter; Positive definite matrix; Metric
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2012.03.002

Most multivariate statistical techniques are based upon the concept of distance. The purpose of this paper is to introduce statistical distance measures, which are normalized Euclidean distance measures, where the covariances of observed correlated measurements x1,…,xn and entries of the Fisher information matrix (FIM) are used as weighting coefficients. The measurements are subject to random fluctuations of different magnitudes and have therefore different variabilities. A rotation of the coordinate system through a chosen angle while keeping the scatter of points given by the data fixed, is therefore considered. It is shown that when the FIM is positive definite, the appropriate statistical distance measure is a metric. In case of a singular FIM, the metric property depends on the rotation angle. The introduced statistical distance measures, are matrix related, and are based on m parameters unlike a statistical distance measure in quantum information, which is also related to the Fisher information and where the information about one parameter in a particular measurement procedure is considered. A transformed FIM of a stationary process as well as the Sylvester resultant matrix are used to ensure the relevance of the appropriate statistical distance measure. The approach used in this paper is such that matrix properties are crucial for ensuring the relevance of the introduced statistical distance measures.